10 seconds approach:
Essentially the question says, there are 5 balls and 6 kids - blue, green, beige, white, yellow, and red.
How can we distribute these 5 balls among these 6 kids.
Mathematically, this will look like
a+b+c+d+e+f = 5
where each variable represent a color and 5 represents the package of poster boards.
Now your task is to divide the 5 balls into 6 different groups.
For example, lets say there are 2 kids and I want to split 5 balls among them.
For this purpose, I will use a stick/partition to divide these 5 balls into 2 groups. Visualise as below:
case-1: OO | OOO
case-2: O | OOOO
Case-3: | OOOO
Case-4: OOOO|O
Case-5: OOO | OOO
Case-6: OOOOO |
So essentially, I am arranging 6 objects in a row of which 5 are identical.
which is equal to 6!/5! = 6
So now, our main question reduces to arranging 10 objects of which 5 each are identical.
Answer = 10!/ 5!5! = 252
This is a question on partition rule in PnC. Once you identify the concept applicable, this should take less than 10 seconds to solve.